Demailly's jet-curvature characterization conjecture for Kobayashi hyperbolicity

Let $X$ be a variety, let $T_X$ be its tangent bundle, and let negative $k$-jet curvature mean that $(X,T_X)$ has a metric satisfying the paper's curvature condition on the $k$-th Demailly–Semple tower. Let $k\in\mathbb{N}$.

Demailly's conjecture. The variety $X$ is Kobayashi hyperbolic if and only if there exists $k\in\mathbb{N}$ such that $(X,T_X)$ has negative $k$-jet curvature.

Negative $k$-jet curvature is described as a stronger, nondegenerate condition implying Kobayashi hyperbolicity, while the converse is motivated by examples showing that the required jet order can be arbitrarily large. The equivalence remains open in the stated generality.